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Solving the Maximum Cardinality Bin Packing Problem with a Weight Annealing-Based Algorithm . Kok-Hua Loh University of Maryland Bruce Golden University of Maryland Edward Wasil American University. 10 th ICS Conference January 2007. Outline of Presentation. Introduction

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solving the maximum cardinality bin packing problem with a weight annealing based algorithm

Solving the Maximum Cardinality Bin Packing Problem with a Weight Annealing-Based Algorithm

Kok-Hua Loh

University of Maryland

Bruce Golden

University of Maryland

Edward Wasil

American University

10th ICS Conference

January 2007

outline of presentation
Outline of Presentation
  • Introduction
  • Concept of Weight Annealing
  • Maximum Cardinality Bin Packing Problem
  • Conclusions

1

weight annealing concept
WeightAnnealing Concept
  • Assigning different weights to different parts of a combinatorial problem to guide computational effort to poorly solved regions.
    • Ninio and Schneider (2005)
    • Elidan et al. (2002)
  • Allowing both uphill and downhill moves to escape from a poor local optimum.
  • Tracking changes in objective function value, as well as how well every region is being solved.
  • Applied to the Traveling Salesman Problem. (Ninio and Schneider 2005)
    • Weight annealing led to mostly better results than simulated annealing.

2

one dimensional bin packing problem 1bp
One-Dimensional Bin Packing Problem (1BP)
  • Pack a set of N = {1, 2, …, n} items, each with size ti, i=1, 2,…, n, into identical bins, each with capacity C.
  • Minimize the number of bins without violating the capacity constraints.
  • Large literature on solving this NP-hard problem.

3

outline of weight annealing algorithm
Outline of Weight Annealing Algorithm
  • Construct an initial solution using first-fit decreasing.
  • Compute and assign weights to items to distort sizes according to the packing solutions of individual bins.
  • Perform local search by swapping items between

all pairs of bins.

  • Carry out re-weighting based on the result of the previous optimization run.
  • Reduce weight distortion according to a cooling schedule.

4

neighborhood search for bin packing problem
Neighborhood Search for Bin Packing Problem
  • From a current solution, obtain the next solution by swapping items between bins with the following objective function (suggested by Fleszar and Hindi 2002).

5

neighborhood search for bin packing problem7
Neighborhood Search for Bin Packing Problem
  • Swap schemes
    • Swap items between two bins.
    • Carry out Swap (1,0), Swap (1,1), Swap (1,2), Swap (2,2) for all pairs of bins.
    • Analogous to 2-Opt and 3-Opt.
  • Swap (1,0) (suggested by Fleszar and Hindi 2002)

Bin α

Bin β

Bin α

Bin β

  • Need to evaluate only the change in the objective function value.

6

neighborhood search for bin packing problem8
Neighborhood Search for Bin Packing Problem
  • Swap (1,1)

( fnew = 164)

(f = 162)

  • Swap (1,2)

( fnew = 164)

(f = 162)

7

weight annealing for bin packing problem
Weight Annealing for Bin PackingProblem
  • Weight of item i

wi = 1 + K ri

  • An item in a not-so-well-packed bin, with large ri,

will haveits size distorted by a large amount.

  • No size distortions for items in fully packed bins.
  • K controls the size distortion, given a fixedri .

8

weight annealing for bin packing problem10
Weight Annealing for Bin Packing Problem
  • Weight annealing allows downhill moves in a maximization

problem.

  • Example C = 200, K= 0.5,

Transformed space f = 70126.3

Original space f = 63325

Transformed space f new = 70132.2

Original space f new= 63125

  • Transformed space -uphill move
  • Original space - downhill move

9

maximum cardinality bin packing problem mcbp
Maximum Cardinality Bin Packing Problem (MCBP)
  • Problem statement
    • Assign a subset of n items with sizes ti to a fixed number of m bins

of identical capacity c.

    • Maximize the number of items assigned.
  • Formulation

10

maximal cardinality bin packing problem
Maximal Cardinality Bin Packing Problem
  • Practical applications
    • Computing.
      • Assign variable-length records to a fixed amount of storage.
      • Maximize the number of records stored in fast memory so as to ensure a minimum access time to the records.
    • Management of real time multi-processors.
      • Maximize the number of completed tasks with varying job durations before a given deadline.
    • Computer design.
      • Designing processors for mainframe computers.
      • Designing the layout of electronic circuits.

11

bounds for maximal cardinality bin packing problem
Bounds for Maximal Cardinality Bin Packing Problem
  • We use the three upper bounds on the optimal number of items developed by Labbé, Laporte, and Martello (2003):
  • We use the two lower bounds on the minimal number of bins developed by Martello and Toth (1990): L2 and L3.

12

outline of weight annealing algorithm wamc
Outline of Weight Annealing Algorithm (WAMC)
  • Arrange items in the order of non-decreasing size.
  • Compute a priori upper bound on the optimal number of items (U*).
    • U* =
    • Update ordered list by removing item i with size ti for which i >U*.

(The optimal solution z* is obtained by selecting the first z* smallest items.)

  • Improve the upper bound U*.
    • Find lower bound on the minimum number of bins required (L3).
    • If L3 > m, reduce U* by 1.
    • Update ordered list by removing item i with size ti for which i >U*.
    • Iterate until L3 = m.
  • Find feasible packing solution for the ordered list with the weight annealing algorithm for 1BP.
  • Output results.

13

solution procedures for mcbp
Solution Procedures for MCBP
  • Enumeration algorithm (LA) by Labbé, Laporte, and Martello (2003)
    • Compute a priori upper bounds.
    • Embed the upper bounds into an enumeration algorithm.
  • Branch-and-price algorithm (BP) by Peeters and Degraeve (2006)
    • Compute a priori and LP upper bounds.
    • Solve the problem with heuristics in a branch-and-price framework.

14

test problems
Test Problems
  • Labbé, Laporte, and Martello (2003)
    • 180 combinations of three parameters
      • number of bins m = 2, 3, 5, 10, 15, 20
      • capacity c = 100, 120, 150, 200, 300, 400, 500, 600, 700, 800
      • size interval [tmin, 99]; tmin = 1, 20, 50
    • For each combination (m, c, tmin), create 10 instances by generating item size ti in the given size interval until
  • Peeters and Degraeve (2006)
    • 270 combinations of three parameters
      • capacity c=1000, 1200, 1500, 2000, 3000, 4000, 5000, 6000, 7000, 8000
      • size interval [tmin, 999]; tmin = 1, 20, 50
      • desired number of items
    • For each combination (m, c, tmin), create 10 instances by generating item size ti in the given size interval until

15

computational results
Computational Results
  • Results on the test problems of Labbé, Laporte, and Martello

(2003)

    • Generated 1800 problems for testing on WAMC .
    • LA and BP used a different set of 1800 problems.
    • Number of instances solved to optimality
      • BP 1800
      • WAMC 1793
      • LA 1759
    • Average running times
      • BP < 0.01 sec (500 MHz Intel Pentium III )
      • WAMC 0.03 sec (3 GHz Intel Pentium IV )
      • LA 3.16 sec (Digital VaxStation 3100)

16

computational results18
Computational Results
  • Generated 2700 problems for testing on WAMC; BP used a

different set of 2700 problems.

  • Computational Results
  • WAMC outperforms BP.
    • BP had difficulties solving instances with
      • Large bin capacities (5000-8000)
      • Large number of items (350-500).
    • WAMC solved all instances with bin capacities
    • WAMC was faster.

17

conclusions
Conclusions
  • WAMC is easy to understand and simple to code.
    • Weight annealing has wide applicability(1BP, 2BP).
  • WAMC produced high-quality solutions to the maximum cardinality bin packing problem.
  • WAMC solved 99% (4458/4500) of the test instances to optimality with an average time of a few tenths of a second.

18